Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (3): 975-1002.doi: 10.1007/s10473-022-0310-x

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Edcarlos D. SILVA, Jefferson S. SILVA   

  1. Federal University of Goiás, Zip code 74001-970, Goiânia, Goiás-GO, Brazil
  • Received:2020-09-09 Revised:2020-10-14 Published:2022-06-24
  • Contact: Edcarlos D. SILVA,
  • Supported by:
    Research was partially supported by CNPq with (429955/2018-9). The first author was partially suported by CNPq (309026/2020-2) and FAPDF with (16809.78.45403.25042017).

Abstract: It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical. The potential V is bounded from below and above by positive constants. Because we are considering a critical term which interacts with higher eigenvalues for the linear problem, we need to apply a linking theorem. Notice that the lack of compactness, which comes from critical problems and the fact that we are working in the whole space, are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems. The main feature in this article is to restore some compact results which are essential in variational methods. Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting. This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.

Key words: Quasilinear Schrödinger equations, linking theorems, superlinear elliptic equations, critical nonlinearities, Bounded potentials

CLC Number: 

  • 35J20